akin j e finite element analysis with error estimators (elsevier, 2005)(isbn 0750667222)(c)(465s) mnfsinhvienzone com

on e nZ ie .C om ne Zo en Vi nh Si SinhVienZone.com https://fb.com/sinhvienzonevn Finite Element Analysis with Error Estimators ne C om An Introduction to the FEM and Adaptive Error Analysis for Engineering Students Si nh Vi en Zo J E Akin AMSTERDAM • BOSTON • HEIDELBERG • PARIS • SAN DIEGO • SAN FRANCISCO • SinhVienZone.com LONDON • SINGAPORE NEW YORK • OXFORD • SYDNEY • TOKYO https://fb.com/sinhvienzonevn Elsevier Butterworth-Heinemann Linacre House, Jordan Hill, Oxford OX2 8DP 30 Corporate Drive, Burlington, MA 01803 First published 2005 Copyright  2005, J.E Akin All rights reserved The right of J.E Akin to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988 ne C om No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London, England W1T 4LP Applications for the copyright holder's written permission to reproduce any part of this publication should be addressed to the publisher Zo Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: permissions@elsevier.co.uk You may also complete your request on-line via the Elsevier homepage (http://www.elsevier.com), by selecting ‘Customer Support’ and then ‘Obtaining Permissions’ en British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library nh ISBN 7506 6722 Vi Library of Congress Cataloguing in Publication Data A catalogue record for this book is available from the Library of Congress Si For information on all Elsevier Butterworth-Heinemann publications visit our website at http://books.elsevier.com Printed and bound in Great Britain SinhVienZone.com https://fb.com/sinhvienzonevn Contents om Preface Notation Introduction 1.1 Finite element methods 1.2 Capabilities of FEA 1.3 Outline of finite element procedures 1.4 Assembly into the system equations 1.5 Error concepts 1.6 Exercises 1.7 Bibliography Mathematical preliminaries 2.1 Introduction 2.2 Linear spaces and norms 2.3 Sobolev norms * 2.4 Dual problems, self-adjointness 2.5 Weighted residuals 2.6 Boundary condition terms 2.7 Adding more unknowns 2.8 Numerical integration 2.9 Integration by parts 2.10 Finite element model problem 2.11 Continuous nodal flux recovery 2.12 A one-dimensional example error analysis 2.13 General boundary condition choices 2.14 General matrix partitions 2.15 Elliptic boundary value problems 2.16 Initial value problems 2.17 Eigen-problems Si nh Vi en Zo ne C SinhVienZone.com https://fb.com/sinhvienzonevn xi xiii 1 12 21 22 24 26 26 28 29 29 31 35 39 39 41 41 56 59 67 69 70 77 80 vi Contents Equivalent forms * Exercises Bibliography 83 86 90 Element interpolation and local coordinates 3.1 Introduction 3.2 Linear interpolation 3.3 Quadratic interpolation 3.4 Lagrange interpolation 3.5 Hermitian interpolation 3.6 Hierarchial interpolation 3.7 Space-time interpolation * 3.8 Nodally exact interpolations * 3.9 Interpolation error * 3.10 Gradient estimates * 3.11 Exercises 3.12 Bibliography 92 92 92 96 97 98 101 106 106 107 110 113 115 One-dimensional integration 4.1 Introduction 4.2 Local coordinate Jacobian 4.3 Exact polynomial integration * 4.4 Numerical integration 4.5 Variable Jacobians 4.6 Exercises 4.7 Bibliography 116 116 116 117 119 123 126 126 Error estimates for elliptic problems 5.1 Introduction 5.2 Error estimates 5.3 Hierarchical error indicator 5.4 Flux balancing error estimates 5.5 Element adaptivity 5.6 H adaptivity 5.7 P adaptivity 5.8 HP adaptivity 5.9 Exercises 5.10 Bibliography 127 127 131 132 136 138 139 139 140 141 143 Si nh Vi en Zo ne C om 2.18 2.19 2.20 SinhVienZone.com https://fb.com/sinhvienzonevn Contents vii Super-convergent patch recovery 6.1 Patch implementation database 6.2 SCP nodal flux averaging 6.3 Computing the SCP element error estimate 6.4 Hessian matrix * 6.5 Exercises 6.6 Bibliography 146 146 158 164 166 176 176 Variational methods 7.1 Introduction 7.2 Structural mechanics 7.3 Finite element analysis 7.4 Continuous elastic bar 7.5 Thermal loads on a bar * 7.6 Reaction flux recovery for an element * 7.7 Heat transfer in a rod 7.8 Element validation * 7.9 Euler’s equations of variational calculus * 7.10 Exercises 7.11 Bibliography 178 178 179 180 185 192 196 199 202 208 210 213 Cylindrical analysis problems 8.1 Introduction 8.2 Heat conduction in a cylinder 8.3 Cylindrical stress analysis 8.4 Exercises 8.4 Bibliography 215 215 215 225 229 229 General interpolation 9.1 Introduction 9.2 Unit coordinate interpolation 9.3 Natural coordinates 9.4 Isoparametric and subparametric elements 9.5 Hierarchical interpolation 9.6 Differential geometry * 9.7 Mass properties * 9.9 Interpolation error * 9.9 Element distortion 9.10 Space-time interpolation * 9.11 Exercises 9.12 Bibliography 231 231 231 238 239 247 252 256 257 258 260 262 263 Si nh Vi en Zo ne C om SinhVienZone.com https://fb.com/sinhvienzonevn viii Contents 265 265 265 267 270 273 276 279 280 11 Scalar fields 11.1 Introduction 11.2 Variational formulation 11.3 Element and boundary matrices 11.4 Linear triangle element 11.5 Linear triangle applications 11.6 Bilinear rectangles * 11.7 General 2-d elements 11.8 Numerically integrated arrays 11.9 Strong diagonal gradient SCP test case 11.10 Orthtropic conduction 11.11 Axisymmetric conduction 11.12 Torsion 11.13 Introduction to linear flows 11.14 Potential flow 11.15 Axisymmetric plasma equilibria * 11.16 Slider bearing lubrication 11.17 Transient scalar fields 11.18 Exercises 11.19 Bibliography 281 281 281 284 289 291 316 318 319 322 337 344 350 358 358 365 370 377 381 382 12 Vector fields 12.1 Introduction 12.2 Displacement based stress analysis 12.3 Planar models 12.4 Matrices for the constant strain triangle 12.5 Stress and strain transformations * 12.6 Axisymmetric solid stress * 12.7 General solid stress * 12.8 Anisotropic materials * 384 384 384 389 395 407 412 413 413 Si nh Vi en Zo ne C om 10 Integration methods 10.1 Introduction 10.2 Unit coordinate integration 10.3 Simplex coordinate integration 10.4 Numerical integration 10.5 Typical source distribution integrals * 10.6 Minimal, optimal, reduced and selected integration * 10.7 Exercises 10.8 Bibliography SinhVienZone.com https://fb.com/sinhvienzonevn Contents ix Circular hole in an infinite plate Dynamics of solids Exercises Bibliography 416 428 435 435 Index 437 Si nh Vi en Zo ne C om 12.9 12.10 12.11 12.11 * Denotes sections or chapters that can be omitted for a first reading or shorter course SinhVienZone.com https://fb.com/sinhvienzonevn om C ne Zo en Vi nh Si SinhVienZone.com https://fb.com/sinhvienzonevn Chapter 12, Vector fields 433 FEA Time−History for Component_2 at Node 60 Numerical Biggs exact 20 om C −20 −40 0.05 0.1 0.15 Time 0.2 0.25 en −60 Zo Newmark Method beta = 1/4, gamma = 1/2 dt = 0.00125 ne Component (max = 47.92, = −58.77) 40 0.3 FEA Time−History for Component_1 at Node Vi nh Si Component (max = 1.311, = −1.355) 0.5 Numerical Biggs exact −0.5 −1 Newmark Method beta = 1/4, gamma = 1/2 dt = 0.00125 0.05 0.1 0.15 Time 0.2 0.25 0.3 Figure 12.37 Newmark method velocity (top), and displacement time history SinhVienZone.com https://fb.com/sinhvienzonevn 434 Finite Element Analysis with Error Estimators FEA Time−History for Component_2 at Node 60 Numerical Biggs exact 20 om −20 C Component (max = 46.83, = −50.82) 40 dt = 0.0025, theta = 1.25 0.05 Zo −60 ne −40 0.1 0.15 Time 0.2 0.25 0.3 en FEA Time−History for Component_1 at Node Vi nh 0.5 Si Component (max = 1.293, = −1.241) Numerical Biggs exact −0.5 dt = 0.0025, theta = 1.25 −1 0.05 0.1 0.15 Time 0.2 0.25 0.3 Figure 12.38 Wilson method velocity (top), and displacement time history SinhVienZone.com https://fb.com/sinhvienzonevn Chapter 12, Vector fields 435 12.11 Exercises Implement the isotropic Ee matrix for the plane strain assumption Assume the stress components are in the xx, yy, xy order Implement the isotropic Ee matrix for the general solid Assume the stress components are in the xx, yy, xy, zz, xz, yz order Give an implementation of the Be matrix for a general solid Assume the strain components are in the xx, yy, xy, zz, xz, yz order Review the scalar implementation for the square matrix for the T3 element, given in Fig 11.6, and extend it to the case of a two-dimensional vector displacement model (that will double the matrix size) Implement a general evaluation of the elasticity Ee matrix that will handle all the 1-, 2-, and 3-dimensional cases Implement a general evaluation of the elasticity Be matrix that will handle all the 1-, 2-, and 3-dimensional cases Modify the plane stress implementation shown in Figs 12.10 and 11 to solve a plane strain problem Remember that an additional normal stress must be considered in the post-processing stage Validate the results with a patch test Use the definition of the kinetic energy of a differential mass, dm, at a point to develop the element kinetic energy and thereby the consistent mass matrix for an element For the three node plane stress triangle use the previous scalar integrals of Eq 11.11 and Fig 11.8 to write, by inspection, the × element mass matrix How much of the mass is associated with the x- and y-directions? Vi en Zo ne C om [2] [3] [4] [5] [6] [7] Akin, J.E., Finite Elements for Analysis and Design, London: Academic Press (1994) Akin, J.E., Object-Oriented Programming Via Fortran 90/95, Cambridge: Cambridge University Press (2003) Babuska, I and Strouboulis, T., The Finite Element Method and its Reliability, Oxford: Oxford University Press (2001) Bathe, K.J., Finite Element Procedures, Englewood Cliffs: Prentice Hall (1996) Becker, E.B., Carey, G.F., and Oden, J.T., Finite Elements − An Introduction, Englewood Cliffs: Prentice Hall (1981) Belytschko, T., Liu, W.K., and Moran, B., Nonlinear Finite Elements for Continua and Structures, New York: John Wiley (2000) Biggs, J.M., Introduction to Structural Dynamics, New York: McGraw-Hill (1964) Si [1] nh 12.12 Bibliography SinhVienZone.com https://fb.com/sinhvienzonevn 436 Finite Element Analysis with Error Estimators [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] om [16] C [14] [15] ne [13] Zo [12] en [11] Vi [10] nh [9] Bonet, J and Wood, R.D., Nonlinear Continuum Mechanics for Finite Element Analysis, Cambridge: Cambridge University Press (1997) Cook, R.D., Malkus, D.S., Plesha, N.E., and Witt, R.J., Concepts and Applications of Finite Element Analysis, New York: John Wiley (2002) Crisfield, M.A., Non-linear Finite Element Analysis of Solids and Structures, Vol 1, Chichester: John Wiley (1997) Desai, C.S and Abel, J.F., Introduction to the Finite Element Method, New York: Van Nostrand - Reinhold (1972) Gupta, K.K and Meek, J.L., Finite Element Multidisciplinary Analysis, Reston: AIAA (2000) Hinton, E and Owen, D.R.J., Finite Element Programming, London: Academic Press (1977) Hughes, T.J.R., The Finite Element Method, Mineola: Dover Publications (2003) Meek, J.L., Computer Methods in Structural Analysis, London: E & F.N Spon (1991) Newmark, N.M., "A Method of Computation for Structural Dynamics," ASCE J Eng Mech Div., 85(EM3), pp 67−94 (July 1959) Oden, J.T., Finite Elements of Nonlinear Continua, New York: McGraw-Hill (1972) Petyt, M., Introduction to Finite Element Vibration Analysis, Cambridge: Cambridge University Press (1998) Segerlind, L.J., Applied Finite Element Analysis, New York: John Wiley (1987) Smith, I.M and Griffiths, D.V., Programming the Finite Element Method, 3rd Edition, Chichester: John Wiley (1998) Stein, E., Error Controlled Adaptive Finite Elements in Solid Mechanics, Chichester: John Wiley (2003) Szabo, B and Babuska, I., Finite Element Analysis, New York: John Wiley (1991) Weaver, W.F., Jr and Johnston, P.R., Finite Elements for Structural Analysis, Englewood Cliffs: Prentice Hall (1984) Wiberg, N.-E and Li, X.D., "A Postprocessed Error Estimate and an Adaptive Procedure for the Semidiscrete Finite Element Method in Dynamic Analysis," Int J Num Meth Eng, 37, pp 3585−3603 (1994) Wiberg, N.-E., "Superconvergent Patch Recovery − A Key to Quality Assessed FE Solutions," Adv Eng Software, 28, pp 85−95 (1997) Zienkiewicz, O.C and Taylor, R.L., The Finite Element Method, 5th Edition, London: Butterworth-Heinemann (2000) Si [8] SinhVienZone.com https://fb.com/sinhvienzonevn Index 364,366 Axisymmetric solid, 6,219,344,377,378 Axisymmetric stress, 225,385,406,412 Aziz, A.K., 89,263 ne C om Babuska, I., 25,90,115,143,144,145,177, 263,264,430 Back substitution, 37 Baker, V.T., 90 Bang, H., 25,383 Bank, R.E., 115 Bar, 81,114,180,352 Barlow points, 109,147,278 Barlow, J., 115,280 Barnhill, R.E., 143 Barycentric coordinates, 267 Bathe, K.J., 24,83,90213,430,435 Beam, 88,113,211 Bearing load, 371,372,373,375 Becker, E.B., 24,263,430 Behr, M., 263 Belytischko, T., 142,175,430 Bending, 404 Biggs, J.M., 430,435 Bilinear form, 28,834 Blacker, T., 143,176 Blended functions, 242 Body forces, 185,189,198,224,389,391, 395,396 Bonet, J 435 Bonnerot, R., 90,264 Bookkeeping, 49,72,182 Boolean assembly, 8,182,183 Boundary conditions, 4,10,36,68,204,287 Boundary flux, 39,53,359,361 Boundary matrices, 4,31,129,284,286,289, 302,318,321,362 Boundary segment, 39,48,75,186,202,284, 289,296,303,307,361,396 Boundary value problem, 71 Brass, 190,197 Brebbia, C.A., 264 Si nh Vi en Zo Abel, J.F., 429 Abdulwahab, F., 144,177 Abramowitz, M., 115,126 Abscissas, 121,122,247,270 Acceleration, 396 Adams, R.A., 90 Adams, V., 24 Adaptivity, 2,29 h-method, 109,139,316,335,349,369, 418 p-method, 103,109,139,249,250 hp-method, 103,139,140 Adjacent elements, 69,149,151,153 Ahmad, S., 213,280,382 Ainsworth, M., 29,90,132,136,140,143, 145,176 Akin, J.E., 24,90,115,176,364,382,430 Algebraic equations, 34,37,40,50,51,70, 72,192,218 Allaire, P.E., 382 Allan, T., 368,382 Allowable error, 172 Angle of twist, 350,351,352,355 Anisotropic material, 1,6,283,320,377,413 Antisymmetry, 286,288,359 APPLICATION _B_MATRIX, 39,355 APPLICATION _E_MATRIX, 39,355 Area, 234,292 Area coordinates, 96,267,275 Askenazi, A., 24 Aspect ratio, 258 Assembly of equations, 3,4,12,13,15,49, 51,72,80,180,182,183,192,193,207,218, 227,293,308,402 Automatic array, 168,195 Average mass matrix, 77,376 Average nodal flux, 4,60,61,68,162,295, 298,328,329,362 Axelsson, O., 90 Axial stress, 9,185,385 Axial stiffness, 180 Axisymmetric integrals, 118,170,320,323, SinhVienZone.com https://fb.com/sinhvienzonevn 438 Finite Element Analysis with Error Estimators Si nh Vi en om Zo ne C function, 98,99 CALC_SCP_ AVE_NODE_FLUX, 158,159 Capacity matrix, 78,201,318,376 Carey, G.F., 24,90,126,263,383 Carpet plot, 297,312,315,348,353,380 Cartesian tensor, 26,27 Carslaw, H.S., 381 Centrifugal load, 186,224 Centroid, 123,125,227,290,395,408 Charafi, A., 25 Charge density, 229 Chung, T.J., 365,382 Ciarlet, P.G., 90,143,264 Circular hole, 416 Class, 356,382 Cohen, M., 177,280 Collocation, 33 Column vector, 49 Complete polynomial, 21,131,158,239,249, 251,275,276,391 Compliance law, 413,415 Conduction, 52,204,215,219,290,293 Conduction matrix, 72,77,216,217,289, 317,344 Conductivity, 68,75 Connectivity, 3,9,12,20,183 Connor, J.C., 264 Consistent mass matrix, 77,374 Consistent loads, 395,396 Consistent source, 52,277,278 Constant Jacobian, 60 Constant strain triangle (CST), 247,394, 400 Constitutive matrix, 157,158,165,169,171, 186,193,224,226,319,387,393,394,399, 404,413 Constitutive transformation, 416 Constraint equations, 37,399 Contains, 356 Continuity: 239 C , 98,112 C , 98,99,100,104,112 C , 99,104 Continuous flux, 10,57,132,165 Contours, 297,300,305,311,314,325,327, 330,332,333,338,340,341,343,368 Convecting edge, 285,287 Convecting face, 285,287 Convection, 52,68,200,203,204,282,283, 292,323 Convection loss, 202,203,205,206,282,309, 313,324 Convergence rate, 21,130,139,334 Cook, R.D., 24,143,176,264,280,381,431 Cookson, R.A., 264 Cooling bar, 80-83 Cooling cylinder, 379 Cooling fin, 306,311 Cools, R., 280 Coon’s function, 251 Coordinate transformation, 117,198,394, 406 Coordinates, Couette flow, 89 Couple, COUNT _ELEMS_ AT _ELEM, 147,149 Craig, A.W., 145 Crank - Nicolson, 77,79,90,264,376 Crisfield, M.A., 435 Cross-derivatives, 175,248,251 Cubic element, 42,45,47,97,99,101,113, 240,241,251,405 Current density, 363,395 Cunningham, J., 264 Current density, 363,395 Curve metric, 124 Cylinder, 215,219,224,225,361 Cylindrical coordinates, 215,344,363 C Brauchli, H.J., 143 Bubble function, 239,248 Buchanan, G.R., 90,213,229 Bulk modulus, 398 Bruch, J.C., 90,106,115 Byrom, T.G., 24,381 SinhVienZone.com Damping matrix, 428 Data, 53,81,92,191,197,204,209,219,228, 293,301,307,312,345,353,378,400,407 Database, 146,153,158 DeBoor, C., 91 Deflection, 417 https://fb.com/sinhvienzonevn Index 439 D_Q_RULE, 273,275 C om E_ AXISYMMETRIC_STRESS, 388 E_ISOTROPIC_STRESS, 406 E_PLANE_STRESS, 388,404 Edge flux, 292,301 Effective stress, 403,405 Effectivity index, 60,130,136 Eigenproblem, 80,396 Eigenvalues, 83 El-Zafrany, A., 264 Elastic modulus, 393 ELASTIC_B_ AXISYMMETRIC, 412 ELASTIC_B_MATRIX, 406 ELASTIC_B_PLANAR, 392,404 Elastic bar, 185,190,196 Elastic modulus, 180,185,193,387 Electrostatics, 229 ELEM_COL_MATRIX, 39 ELEM_COORD, 170 ELEM_SQ_MATRIX, 39,43,44,46,52,56, 194,203,217,290,320,355,366,371,404 Element based patch, 59,146,153 Element connectivity, 5,17,20 coordinates, 52,194,242,255 error, 21,66,164,258,418 error energy, 171 flux, 68,298 incidences, 18,147 interface, 236,360 matrices, 284,289,293,317,345,355, 364,365,370,372,377,391,398,401 neighbors, 160 properties, 373 reactions, 191,197,199,205 size, 3,22,50,109,130,139,141,300, 314,315,342,349,362,418 strain, 192,193,208,209,383,393,402 stress, 128,192,197,226,227,383, 390,403 type, 155,169,303,307 ELEMENT _NODES, 147,148,149,150,151 Elliptic problem, 127 Encapsulated, 351,356 Energy norm, 134 Si nh Vi en Zo ne Degree of freedom number, 12,18,155,183 Demkowicz, L., 90,383 De-refinement, 139,140 Derivative: global, 57 local, 56,94,165,224,243,244,245,246, 262 nodal parameters, 100 orders, 31 second, 99,153,166 DERIV _C1_L, 100 DERIV _3_L, 98 DERIV _3_T , 233 DERIV _4_Q, 235 DERIV 2_C1_L, 100 Desai, C.S., 24,79,90,115,381,431 DETERMINE_SCP_BOUNDS, 160 Dettmer, W., 106,115,264 Devloo, P., 382 DeVries, G., 24 Diagonal mass matrix, 77,376 DIAGONALIZE_SQ_MATRIX, 77 Differential geometry, 252,319 Differential operator, 7,30,39,127,383,384 Diffusivity, 79 Dirac delta distribution, 33 Dirichlet condition, 10,69,70,131,286, 326,344,361,374 Direct access, 153,155 Direct assembly, 12,400 Direction cosines, 198 Displacement vector, 179,383,384,391,396 Displacements, 3,5,180,390 Distortion, 258 Displacement vector, 179,383,384,391, 396,402,417 Distortional energy, 399 Divergence theorem, 8,27 Dual problem, 30,31 Dunavant, D.A., 271,280 Dunavant quadrature, 271,275 DUNAVANT _UNIT _TRIANGLE, 271 Dynamic memory, 160 Dynamic system, 396 Dym, C.L., 214 SinhVienZone.com https://fb.com/sinhvienzonevn 440 Finite Element Analysis with Error Estimators C om Flow chart, Flux averaging, 39,52,61,158,162,221,223 Flux balance, 136 Flux components, 127 Flux error, 62,64,171 Flux norm, 171 Flux reaction, 73,223 Flux recovery, 57 Flux vector, 298,310,311,322 Force vector, 193,224,328,329,339 FORM_ELEMS_ AT _EL, 151 FORM_L_ ADJACENT _NODES, 148 Forces, 6,179,185,190,199,224,390, FORCE_ AT _TIME, 430 391,395,396 Forward difference, 77,375 Forward substitution, 37 Fourier law, 69,128 Fourier number, 79 Fracture mechanics, 260 Functional analysis, 29 Functional, 11,371 Fundamental magnitudes, 253,254,255 Si nh Vi en Zo ne Energy norm error, 66,222,223,300,310, 330,332,343,362,425 Enrichment, 251 Equation numbers, 12,20,183 Equation of motion, 396 Equilibrium, 163,179,180,216,389,402 Equivalent form, 83 Error density, 138,173 Error estimates, 2,5,29,39,60,79,127,131, 164,334,367 Error energy norm, 12,14,129,166 Estimated error, 427 Essential boundary condition, 3,10,11,30, 31,32,37,51,55,69,72,79,80,127,192,218, 227,293,414,420 Euler theorem, 8,12,208,282 Euler integration, 375 EVAL_SCP_FIT _ AT _PATCH_NODES, 162,175 Exact displacement, 407,408,420 Exact error, 171,342 Exact flux, 60,61,64,65,142 Exact flux error, 67 Exact integrals, 40,52,117,265,271 Exact solution, 47,53,54,61,73,82,141,154, 170,203,205,218,327,333,346,348 EXACT _SOLUTION , 39 Exact source, 376 EXACT _SOURCE, 39 Exact stress, 408,409,421,428 Exploded mesh, 305,412,424,425 Face based patch, 59,146,153 Face convection, 287,292,302,306 Face nodes, 153 Factorization, 36,37 Failure criterion, 5,228,394,405 Ferrari, R.L., 25,264,382 FILL_TYPE_INTERPOLATIONS, 169 Film thickness, 370,371 Fin, 306-316 First fundamental form, 253 Fix, G.J., 90 Flannery, B.P., 177 Flick’s law, 69 Flow around cylinder, 361 SinhVienZone.com Gago, J., 145 Galerkin criterion, 8,28,35,40,41,43,45, 46,72,178,390 Galerkin in time, 77,375 Gallagher, R.H., 176,432 Gamma function, 266 Ganjoo, D.K., 90,115,264 Gardner, G.A., 264 Gardner, L.R.T., 264 Gartling, D., 90,214 Gather, 8,9,50,55,155,165,170,180,184 Gauss’ Theorem, 27 GAUSS_COEFF, 274 GAUSS_2D, 273,276 GAUSS_3D, 273,274 Gaussian quadrature, 40,56,120,121,276 Gellert, M., 280 Generalized trapezoidal integration, 375 Geometric interpolation, 242,244,246,269 Geometric matrix, 94,234 Geometric parameters, 290,292,309 Geometric properties, 256,347 https://fb.com/sinhvienzonevn Index 441 C om Harbord, R., 280 Haroun, M., 280 Hayashi, H., 368,382 Heat balance, 310 Heat conduction, 1,127,199,215,291,337 Heat convection, 68,70,200,203,206,206 Heat generation, 127,216,218,289 Heat flux, 5,79,128,293,299,340,341 Heat loss, 206,282,309,324 Heat transfer, 6,69,283,293 Heinrich, J.C., 90 Hermite interpolation, 98,99,100,104 Hessian, 166 Hexahedra: elements, 96,239 interpolation, 241 Hierarchical interpolation, 101,102,132, 241,246 Hilbert space, 84 Hinton, E., 126,280,381 Hole in infinite plate, 416 Hooke’s law, 69,128,186,194 Hoop strain, 223,226,411 Hoop stress, 227,411 Hu, K.-K., 264 Huang, H.C., 79,91,139,143,176,381 Huebner, K.H., 24,368,381 Hughes, T.J.R., 24,29,91,125,175,264,280, 375378,,383,432,436 Hyperbolic functions, 108 Si nh Vi en Zo ne GET _DLH_ AT _QP, 41,56,76,170,203,217, 224,320,321,323,355,365,376,404 GET _DOF_INDEX, 19 GET _ELEM_DOF, 154 GET _ELEM_INDEX, 14,17,154 GET _ELEM_NODES, 17,154,170 GET _ELEM_QUADRATURES, 169 GET _ELEM_SHAPE, 161 GET _ELEM_TYPE_DATA, 161,169 GET _G_ AT _QP, 41,43,44,404 GET _H_ AT _QP, 41,56,76,170,194,203, 217,224,320,321,323,355,365,377,404 GET _INDEX_ AT _PT , 14,16 GET _LT _FACES, 152 GET _PATCH_QUADRATURE_ORDER, 169 GET _REACTIONS, 74 GET _REAL_LP, 76,194,203,206,217,224, 290,321,355,370,377,387 GET _REAL_MISC, 203,206,355,365,370, 435 GET _REAL_MX, 323,324 GET _SCP_PT _ AT _XYZ, 162 Global approximation, 40,41,43,44,45,46,47 Global array, 43,56 Global coordinates, 53,81,92,191,197,204, 209,219,228,293,301,307,312,345,353, 378,402,409 Global derivatives, 33,42,44,47,49,51,56, 165,244,246 Global error, 173 Global variable, 43 Gradient, 55,127,165,246,247,325 Gradient estimates, 110 Graph, 46,54,60,62,380,433 Gravity, 186,190,198 Green’s theorem, 28,74,75 Gresho, P.M., 176 Griebel, M., 263 Griffiths, D.V., 25,126,432 Gupta, K.K., 24,432 H norm , 30 H error, 171 H norm, 171,172,174 Half symmetry, 212,288,301,305,369 Hansbo, P., 264 SinhVienZone.com Idesman, A., 264 Ill-condition, 60 Include file, 4,13,81 Incompressibility, 226,278,386,399 Inertia tensor, 256 Influence domain, 49 Initial condition, 78 Initial strain, 191,195,389,393 Initial stress, 389 Initial value problem, 74,77,375 Inner product, 29,30,31,129 Integrate by parts, 8,28,30,31,42,43,85 Integration, 116,265 Interface, 39,41,149,151,156,159,167,194, 195 https://fb.com/sinhvienzonevn 442 Finite Element Analysis with Error Estimators Vi C ne en Zo Jacobi iteration, 83 Jacobian, 60,94,116,125,162,243,257,258, 259,262,269,316 Jacobian determinant, 245,246 Jacobian inverse, 245,257 Jaeger, J.C., 382 Jamet, P., 90,264 Johnston, P.R., 214,230,432 Jump term, 136,139 Lame parameters, 254 Laplace equation, 211 Laplacian, 128 Least squares, 34,40,41,44,45,57,158,161 Least squares in time, 77,375 Legendre polynomials, 103,105,250 Leguillon, D., 144 Li, X.D., 432,436 Linear hexahedra, 158,239,241 line, 49,50,52,54,72,92,95,96,133,188, 201,202,216,220,370,371 quadrilateral, 237,243 spring, 179 tetrahedron, 95,107,237,268 triangle, 95,107,237,245,247,257,266, 269,289,291,301,304,314,379 Linear space, 29 LIST _ELEM_FLUXES, 154,156,158,161 LIST _ELEM_ AD_EXACT _FLUXES, 154 LIST _ELEM_TORSION _STRESS, 141, 356,357 LIST _ELEM_TORSION _INTEGRAL, 356, 358 Liu, W.K., 432 Liusternik, L.A., 29,91 Lobatto rule, 122 Local coordinates, 92,231 Local derivatives, 55,94,165,224,243, 244,245,246,262 Locking, 399 Load vector, Loubignac, G., 163,177 Lubrication, 368 Lumped matrices, 50 om Interior residual, 134,136 Interpolate solution, 73 Interpolation error, 107,257 Interpolation functions, 9,48,50,57,92,125, 165,186,226,231,241,242,317,384 Initial strain, 389,391 Initial stress, 389 Inverse Jacobian, 245 INVERT _JACOBIAN , 170,217,223,320, 356,365,376 INVERT _SMALL_MAT , 169 INVERT _2BY 2, 404 Inviscid flow, 364 Irons, B.M., 202,213,280,382 Isoparametric element, 94,95,232,236,242, 253,318,404 Iterative solution, 76 Si nh Keast, B., 272,280 KEAST _UNIT _TET _RULE, 272 Kelly, D.W., 143,144 Keyword, 41,53,153,193,198,206,208,219, 224,228,293,296,301,303,304,307,321, 346,351,372,374,375,378,400,420 Kimser, P.G., 264 Kreyszig, E., 346,383 Krishnamoorthy, C.S., 115 Krizek, M., 144,176 Kundu, T., 24,115,382 Kwon, Y.W., 25,383 L norm, 30,166,170,171 Ladeveze, D., 144 Lagrange interpolation, 11,13,97,239, 268,318 Lakhany, A.M., 177 Lame constants, 399 SinhVienZone.com Maddox, J.R., 176 Magnetic flux density, 363,396 Malkus, D.S., 24,143,176,264,280,382 Martin, H.C., 383 Mass damping, 430 Mass matrix, 76,80,218,319,375,376,428 Mass properties, 256 Material axes, 394,409,415,416 Material property, 5,9,187,190,204,216,228, 291,293,306,312,346,352,373,398,400, https://fb.com/sinhvienzonevn Index 443 ne C om boundary condition code, 53,81,191, 197,204,209,219,400 constraints, 10,38 coordinates, 53,81,92,191,197,204,209 displacements, 5,386,417 exact, 106 forces, influence domain, 49 moment, parameters, 9,71,98,181 pressure, 368 properties, 370 temperature, 5,3,68,79,127,201,205, 206,219,220,283,293,297,299,303, 305,315,336,337,348 thickness, 370 velocity, 362 Nodally exact solution, 71,106 Non-conforming element, 236,360 Norm, 29,30,68,165 Normal flux, 282 Normal stress, 387,390,421,422 Normal vector, 11 Norrie, D.H., 24 Nowinski, J.L., 29,90 Numerical integration, 40,57,119,162,165, 202,203,217,224,227,247,270,317,319, 320,365,376,404 Si nh Vi en Zo 407 Matrix inverse, 36,71,72 Matrix partition, 70 Maximum shear stress, 350,354 Measure, 30,258 Mechanical work, 179,185,384 Meek, J.L., 24,383 Meier, D.L., 91 Mesh adaptivity, 4,419,424 Method of moments, 35 Method of Weighted Residuals (MWR), 32-36,71-78,86 Metric matrix, 253,255 M.H.D plasma, 363 Mid-edge nodes, 239,240,268 Minimal integration, 276 Minimization, 34 Mininum total potential energy, 84,179 Mitchell, A.R., 90,214 Mixed boundary condition, 10,39,70,282, 302,312 Mixed condition matrices, 292,309,323 MIXED_SQ_MATRIX, 39,292,323 Moan, T., 280 Mohr’s circle, 406,409,416 Moment, Monk, P., 263 Moran, B., 436 Morgan, K., 90,133,144,264 Multiple point constraint (MPC), 10,38, 213 Myers, G.E., 200,213,383 Natural boundary conditions, 10,11,31,70, 210,284,286,359 Natural coordinates, 94,97,120,121,200, 234,238,241,273 Natural norm, 30,129 Neighbor lists, 147,152 Neittaanmaki, P., 144,176 Neumann condition, 10,70,131,321,375 Newmark, N.M., 429,436 NEWMARK _METHOD_SYM, 429 Niekamp, R., 264 Nodal based patch, 59,146,153 SinhVienZone.com Oden, J.T., 24,28,90,91,126,132,136,140, 143,144,176,263,383,436 One-eighth symmetry, 293,297,349,352, 376,380 Optimal points, 110,112,279 Orthogonal functions, 29,103,132 Orthotropic material, 286,336,337,415 Outer product, 43,46 OUTER_PRODUCT , 76 Overlapping patches, 164 Owen, D.R.J., 126,381,431 Parallel axis theorem, 256 Parametric curve, 252,254 Parametric derivative, 56,94,165,174,224, 243,244,245,246,262 Parametric equation, 123,252 https://fb.com/sinhvienzonevn 444 Finite Element Analysis with Error Estimators POST _PROCESS_GRADS, 154,155 POST _PROCESS_MIXED, 309 Potential energy, 179,185 Potential flow, 5,356,359,362,363,364 Press, W.H., 177 Pressure, 227,228,371,373 Principal stresses, 410 C om Quadratic hexahedra, 241 hierarchical, 118 line, 42,47,97,98,111,117,219,228,258 quadrilateral, 12,110,241,313,405 tetrahedra, 238,268 triangle, 110,238,266,326,345,352,405, 414 Quadratic functional, 181,210 Quadratures, 40,43,44,55,76,271,272 Quadrature order, 164 Quadrature point, 58,60,113,153,157,354, 362,408 Quadrilateral element, 12,110,234,237, 241,243,313,405 Quarter point element, 260 Quarter symmetry, 22,288,336 Si nh Vi en Zo ne Parametric surface, 6,253 PARM_GEOM_METRIC, 255,319,321,323 Pascal triangle, 238 Patch bounds, 157,158 degree, 153,158 domain, 58,59 element based, 59,146,153 face based, 59,146,153 flux, 39,52,60,61,158,162,221,223 interpolation, 57,59,63,163 Jacobian, 157 node based, 58,59,146,153 overlapping, 164 parametric space, 60 type, 169 Patch test, 202,206,209,359,407 Pepper, D.W., 90 Peric, D., 106,115,264 Petyt, M., 428,436 Piecewise approximation, 2,49,55,171 Pironneau, O., 25 Pitkaranta, J., 115,263 Planar elasticity, 128 Plane strain, 384,388,390 Plane stress, 384,388,390,393,400,404,407, 414 Plasma, 365 Plesha, N.E., 24,143,176,264,382 Plot, 42 Point load, 185,390 Point source, 303 Poisson equation, 211,212,228,320,326, 350,355,359 POISSON _ ANISOTROPIC_E_MATRIX, 320 Poisson’s ratio(s), 387,413 Polynomial degree, 140 Portela, A., 25 Position vector, 252 Positive - definiteness, 11,29 Post-process, 8,10,39,50,192,307,356,389 Post-solution calculations, 190,323,372 POST _PROCESS_ELEM, 39,195,206, 324,356,372 SinhVienZone.com Radial displacement, 223 Radial position, 217,223 Random access, 153,155 Rao, S.S., 26 Razzaque, A., 213,382 Reaction flux, 53,80,196,219,223,308 Reaction force, 192,193,403 Reactions, 6,10,51,53,71,73,190,191,196, 205,209,293 Record number, 153,155,157 Rectangular element, 107,316 Reddi, M.M., 368,383 Reddy, J.N., 90,214 Reduced integration, 276 Refinement, 139,335 Refinement indicator, 138,139,174,335 REAL_IDENTITY , 365 Reshape, 52 Residual error, 7,8,32,33 Reynolds equation, 368,371 https://fb.com/sinhvienzonevn Index 445 Shear stress, 350,353,387,390,423 Shephard, M.S., 383 Silvester, P.P., 25,264,382 Simple harmonic motion, 396 Simplex elements, 95,107,131,232,237,261 Single element solution, 36,43,44,45,46, 202 Singular Jacobian, 258,259 Singular value decomposition, 162 Scalar field, 281 Skyline, 429 Scatter, 8,9,21,50,153,180,391 SKY _TYPE_1_SYM, 430 Schwartz inequality, 28,29 Slider bearing, 368,371 SCP averages, 68,347,354,362 Slope, 98 SCP recovery, 57,132,146,322 Smith, I.M., 25,126 SCP_ERROR_ESTIMATES, 165,167,175 Soap film, 350 Second derivatives, 153,166 Sobolev norm, 30 Secrest, D., 126 Soblov, V.J., 90 Segerlind, L.J., 25,264,351,383 Solution error, 32,353 SEG_COL_MATRIX, 292,321 Solution integral, 351,355,358 Selective integration, 276 Solution techniques, 36 SELECT _ APPLICATION _B_MATRIX, 170 Solution vector, 33 SELECT _ APPLICATION _E_MATRIX, 169 Source, 9,32,42,48,75,76,283,293,376 SELECT _EXACT _FLUX, 171 Source integrals, 72,77,273,365 SELECT _EXACT _SOLUTION , 170,320 Source vector, 33,48,51,129,201,216 SELECT _EXACT _SOURCE, 376 Source resultant, 53,72,290,376 Self-adjoint, 30,365 Space-time interpolation, 78,106,260 Semi-discrete, 77,78 Space-time slab, 106,107,261 Semi-infinite element, 108 Specific heat, 79 Semi-norm, 30 Sphere, 346,383 Sequential data, 153,155 Spring, 179 Serendipity elements, 131,239,240 Spring-mass system, 432 SET _ELEM_TYPE_INFO, 154 Square matrix, 9,48,50,55,72,77,129 Shames, I.H., 214 Steady state, 80,284,375 SHAPE_3_L, 98 Steel, 190,197 SHAPE_3_T , 232 Stegun, I.A., 115,126 SHAPE_4_12_Q, 240 Stein, E., 436 SHAPE_4_Q, 235,241 Steinberg, R., 144,176 SHAPE_6_T , 237 Step bearing, 372 SHAPE_8_H, 241 Stiffness damping, 430 SHAPE_8_Q, 241 Stiffness matrix, 9,33,180,187,189,194, SHAPE_16_Q, 241 198,201,224,391,398,401 SHAPE_20_H, 241 STORE_COLUMN , 16,17 SHAPE_32_H, 241 STORE_FLUX_POINT _COUNT , 52,56,217 SHAPE_C1_L, 100 STORE_FLUX_POINT _DATA, 52,56,217 Shear modulus, 350,393,399,413 STORE_FULL_SQUARE, 16,17 Shear strain, 390 Si nh Vi en Zo ne C om Rigid body motion, 227,278 Robin boundary condition, 10,39,70,213, 282,284,302,323,374 Robinson, J., 203,214 Rockey, K.C., 229 Rod, 199 Ross, C.T.F., 229 Rossettos, J.N., 115 SinhVienZone.com https://fb.com/sinhvienzonevn 446 Finite Element Analysis with Error Estimators ne C om 336,337,348 Test function, 74 Tetrahedra, 95,237,238,261 Teukolsky, S A., 177 Tezduyar, T.E., 91,115,264 Thermal conductivity, 3,9,12,20,183,216 Thermal diffusivity, 79 Thermal expansion, 192 Thermal load, 193,197 Thermal strain, 192,194,197,394,395 Thickness, 283,290,320,390,395 Thornton, E.A., 24,382 Time derivative, 74,75,78,79,281,376 Time history, 80,82,83,375,377,377 Time slab, 78,261,263 Time step, 79,376 Tokamak, 363 Tong, P., 106,115 Torque, 350 Torsion, 349,351,352,353,354,355 Total potential energy, 11,187,384 Touzot, G., 177 Tractions, 128,129,185,187,390,391,396 Transformation matrix, 394,406,409,416 Transient applications, 74,75,76,79,106, 281,377-380 Transition element, 239 Trapezoidal rule, 119,120 Triangular elements, 22,95,107,110,237, 245,247,257,266,267,271,326,345,352, 405,407,414 Triangle inequality, 28,30 Si nh Vi en Zo Strain, 192,193,208,209,384,393 Strain energy, 171,179,185,384 Strain energy norm, 68,138,166 Strain-displacement relation, 165,171,186, 188,192,226,384,386,392,393,398,412 Strain transformation, 394,406,410,416 Stream function, 356,363,368 Stein, E., 264 Stress, 128,192,197,226,227,384,390 Stress concentration, 424 Stress error, 165 Stress error norm, 130 Stress free temperature, 394 Stress function, 350,353 Stress-strain law, 392 Stress transformation, 406,409 Strang, W.G., 90 Strong form, 83 Strong typing, 41,233 Strouboulis, T., 90,143,383 Stroud, A.H., 126 Structural mechanics, 179 Subdomain method, 36 Sub-parametric elements, 242,247 Subset, 49,186 Super convergence patch (SCP), 57,58,81 Super-parametric elements, 242 Surface gradient, 254 Surface metric, 254 Surface normal, 253 Surface tractions, 390 SVDC_BACK _SUBST , 162 SVDC_FACTOR, 162 Swartz, S.E., 264 Symmetry, 22,29,286,293,359 SYMRUL, 270 System equations, 17 System norm, 167 Szabo, B., 25,90,115,144,177,263 Tangent vectors, 252,254 Tangential derivatives, 101,248 Taylor, R.L., 25,91,115,126,177,214,383 Taylor series, 107,176,257 Temperature, 3,68,79,127,201,205,206, 219,220,283,293,297,299,303,305,315, SinhVienZone.com Union, 75 Unique solution, 86 Unit coordinates, 94,95,97,120,121,231, 237,265 Unit normal vector, 74,128 Unit tetrahedra, 272 Unit triangle, 267,271 Unstructured mesh, 153 Unsymmetric equations, 34,36 Upadhyay, C.S., 143 UPDATE_SCP_STATUS, 157 Usmani, A.S., 79,91,139,143,176,383 https://fb.com/sinhvienzonevn Index 447 Weight, 194 Weighted residual, 7,32,42 Weighting function, 33 Weights, 121,122,247,270,271,272 Wiberg, N.-E., 144,177,432,436 Witt, R.J., 24,143,176,264,381 Whiteman, J.R., 25,91,143,176 Wilson method, 434 Wood, R.D., 435 Wooten, J.W., 364,383 Work, 179,185,384 om Yield stress, 405 Young’s modulus, 387 ne C Z-Z error estimator, 57,60,139 Zhu, J.Z., 60,91,144,145,177,383 Zienkiewicz, O.C., 25,60,91,115,126,133, 144,145,177,214,230,264,280,383 Ziukas, S., 144,177 Zyvoloski, G., 90,106,115,144 Si nh Vi en Zo Validation, 202 Variational form, 7,11,83,178,200,216, 281,364 Vector field, 383 Vector plot, 298,310,311,328,329,339, 354,417 Vector potential, 363 VECTOR_MULT _SKY _SYM, 430 Velocity potential, 356,359 Velocity, 358,359,363,370 Vettering, W.T., 177 Viscosity, 370 Volume, 166,170,189,408 Volume coordinates, 96,268 Von Mises stress, 228,405,411,419,428 Wada, S., 368,383 Wait, R., 90,214 Weak form, 8,53 Weaver, W.F., 214,230,432 Wedge element, 107 SinhVienZone.com https://fb.com/sinhvienzonevn ... Number of geometric nodes per element Maximum number of nodes for element type Dimension of parametric space for element type Number of quadrature points for element type Current element type shape... of element geometry nodes Number of generalized parameters (dof) per node Number of flux components per segment node Number of integer element properties Number of floating point (real) element. .. numbers connected to a particular element is usually referred to as the element incident list for that element We usually assign a material code, or properties, to each element Finite element analysis